The proof's quite clear, but what if I have been asked to evaluate 10^log 3? Do I just use the law without having to prove anything? Would I still get the same marks in a test compared to someone who can explicitly prove it?

One can either first define and and the define and as their inverses or vice-versa. Either way , , , and follow from the definition of "inverse" function:
g(x) is the inverse function to f(x) if and only if both f(g(x))= x and g(f(x))= x.